PHYSICAL REVIEW E 84, 011148 (2011)
Perturbative path-integral study of active- and passive-tracer diffusion in fluctuating fields
Vincent Démery and David S. Dean
Laboratoire de Physique Théorique, IRSAMC, Université Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 4, France
(Received 8 April 2011; published 29 July 2011)
We study the effective diffusion constant of a Brownian particle linearly coupled to a thermally fluctuating
scalar field. We use a path-integral method to compute the effective diffusion coefficient perturbatively to lowest
order in the coupling constant. This method can be applied to cases where the field is affected by the particle
(an active tracer) and cases where the tracer is passive. Our results are applicable to a wide range of physical
problems, from a protein diffusing in a membrane to the dispersion of a passive tracer in a random potential. In
the case of passive diffusion in a scalar field, we show that the coupling to the field can, in some cases, speed up
the diffusion corresponding to a form of stochastic resonance. Our results on passive diffusion are also confirmed
via a perturbative calculation of the probability density function of the particle in a Fokker-Planck formulation
of the problem. Numerical simulations on simplified systems corroborate our results.
DOI: 10.1103/PhysRevE.84.011148
PACS number(s): 05.40.−a, 05.60.Cd
I. INTRODUCTION
Diffusion in a quenched random medium is a problem
which has been extensively investigated [1,2]. An important
physical problem is to understand how the long-time transport properties of a Brownian particle, notably its effective
diffusion constant, are modified with respect to those of a
homogeneous medium. Two types of problems with quenched
disorder have been extensively studied. The first is diffusion
in a medium where the local diffusion constant depends
on spatial position and is taken to have some statistical
distribution, and the second is for a particle that is advected
by a quenched random velocity field. The case where the
random velocity field is derived from a potential is of
importance as a toy model for spin glasses and glasses as
it can exhibit a static spin glasslike transition [3,4] and a
dynamical structural glasslike transition [5–7]. Variants of
the toy model of diffusion in a random potential can exhibit
a transition in their transport properties, notably diffusion,
which is normal in the high-temperature phase and becomes
subdiffusive in the low-temperature phase [1,8–10]. The onset
of the subdiffusive regime is signaled by the vanishing of
the late-time diffusion constant. It should also be mentioned
that the above class of problems can also be related the
problem of evaluating the macroscopic (effective) electrical
properties, such as conductivities or dielectric constants, of
random conductors or dielectrics [2].
The case of diffusion in dynamically evolving random
media has perhaps received less attention. The most widely
studied problem of a time-dependent nature is for a particle
diffusing in a turbulent flow [11,12], and the particle here is a
passive tracer and has no effect on the flow field. Another
example is for a protein diffusing in a membrane, where
the protein is subject to a force generated by membrane
curvature or composition, and these quantities themselves
fluctuate. However, in this case the membrane is in general
affected by the protein [13]. Naively one might expect that
protein diffusion is speeded up by coupling to height or
composition fluctuations in a membrane; however, this is
not the case: The feedback of the protein on the membrane
configuration actually slows the protein diffusion with respect
1539-3755/2011/84(1)/011148(11)
to a nonfluctuating homogeneous one. A general question
that arises in these sorts of problems is: When does the
fluctuating field speed up the diffusion of a tracer and when
does it slow it down? An important question, which has
received much recent attention, is how the diffusion constant
of a protein depends on its size. The classic hydrodynamic
computation of Saffmann and Delbrück [14] treats the protein
as a solid cylinder in an incompressible layer of fluid (the lipid
bilayer) sandwiched between another fluid (the water). In this
formalism the protein diffusivity shows a weak logarithmic
dependence on the cylinder radius. However, the validity of
this result has been called into question experimentally where
a stronger dependence is reported [15]. A number of theoretical
studies have suggested that a protein’s diffusion could be
modified by coupling to local membrane properties, such as
composition and curvature, that are not taken into account in a
purely hydrodynamic model for a membrane [13,16–22]. One
should bear in mind that protein coupling to local geometry
and composition has also been postulated as a mechanism for
interprotein interactions in biological membranes [23,24].
In this paper we will consider a particle, whose position
is denoted by x(t), diffusing in a scalar field hψ, where h
is a parameter controlling the coupling of the field to the
particle. Thus the particle drift is given by u = −κ x h∇ψ,
where κ x denotes the bare diffusion constant of the particle.
The dynamics of both the particle and the field are overdamped
and stochastic. Our aim is to evaluate how the effective
diffusion coefficient for the particle is modified by its coupling
to the field. A path-integral approach allows us to perform a
perturbative computation if the coupling constant between the
particle and the field is small for a wide range of physically
different problems. Within this formalism we recover some
previous results on the diffusion of active tracers where the
whole system (particle plus field) obeys detailed balance [22].
However, the method also allows us to analyze passive
diffusion and a continuum of intermediate models with varying
degrees of feedback of the particle on the field and for thermal
fluctuations on the particle and field that are not necessarily
at the same temperature. This intermediate range of models
could apply to the study of tracers in nonthermally driven
fields and have applications to active colloidal systems [25],
011148-1
©2011 American Physical Society
VINCENT DÉMERY AND DAVID S. DEAN
PHYSICAL REVIEW E 84, 011148 (2011)
swimmers [26], or systems where the tracer is heated with an
external heat source such as a laser [27]. The basic method
may also be useful for studying systems where the field itself
is nonthermally driven, for instance, lipid bilayers, where the
field is driven by an electric field [28].
The formalism also allows us to see how the relative
time scales between the tracer and fluctuating field affect the
effective diffusion constant of the tracer.
Our results for the effective diffusion constant show very
different effects for active and passive diffusion, and diffusion
in a quenched potential. In pure active diffusion, where the
stochastic equations of motion obey detailed balance, the
diffusion is always slowed. For passive diffusion, if the field
evolution is slow, the particle slows, as if it was evolving in a
quenched potential [2]. As the field evolution is speeded up, the
diffusion constant increases and then reaches a maximum, in
a manner reminiscent of stochastic resonance [29–32]. As the
field evolution rate increases still further, the diffusion constant
diminishes and eventually reaches the bare value it would
have in the absence of coupling to a field. The results derived
via the path-integral formalism for this case are also derived
via perturbation theory on the corresponding time-dependent
Fokker-Planck equation.
Finally we demonstrate the analytically predicted effects by
numerical simulation of simple models where the fluctuating
field has a small number of Fourier modes.
II. THE MODEL
To start, we define the general class of models of interest
to us in this paper. First, we consider the dynamics of a
Langevin particle whose position is denoted by x(t) diffusing
in a d-dimensional space with a linear coupling to a fluctuating
Gaussian field, as introduced in Ref. [22]. The overall energy
(Hamiltonian) for the system is
1
H =
2
φ(x)φ(x) d x − hKφ[x(t)],
(1)
where the first term is the quadratic energy of a free scalar
field, and the second corresponds to the tracer seeing an
effective potential hψ = −hKφ. We will take and K to be
self-adjoint operators. For two operators A(x, y) and B(x, y)
we will denote by AB(x, y) their composition as operators:
(AB)(x, y) = A(x,z)B(z, y) d z.
For a system obeying detailed balance (and whose equilibrium state is thus given by the Gibbs-Boltzmann distribution)
the particle evolves according to
δH √
+ κ x η(t)
δx
√
= hκ x ∇Kφ[x(t)] + κ x η(t),
ẋ(t) = −κ x
(2)
where the noise term is Gaussian with mean zero and
correlation function
η(t)η(s)T = 2T δ(t − s)1,
where T is the temperature of the system.
(3)
In the absence of a coupling between the field and the
particle (h = 0), the particle diffuses normally, and the meansquared displacement at large times behaves as
[x(t) − x(0)]2 ∼ 2dT κ x t = 2dD x t,
t→∞
(4)
where d is the spatial dimension and D x = T κ x is the bare
diffusion constant.
We take a general dissipative dynamics for the field [33]:
δH
√
+ κφ ξ (x,t)
δφ(x)
√
= −κφ Rφ(x) + hκφ RK[x − x(t)] + κφ ξ (x,t),
(5)
φ̇(x,t) = −κφ R
where R is a self-adjoint dynamical operator and ξ is a
Gaussian noise of zero mean that is uncorrelated in time.
In order for the field to equilibrate to the Gibbs-Boltzmann
distribution, the correlation function of this noise must be
ξ (x,t)ξ ( y,s) = 2T R(x − y)δ(t − s).
(6)
The effective diffusion constant for the particle is defined
via
[x(t) − x(0)]2 ∼ 2dT κe t = 2dDe t,
t→∞
(7)
where De is the effective or late time diffusion constant.
Our model applies to many systems: A point magnetic
field of magnitude h diffusing in a Gaussian ferromagnet can
be modeled with (x, y) = (−∇ x2 + m2 )δ(x − y), K(x, y) =
δ(x − y), and R(x, y) = δ(x − y) for model A dynamics or
R(x, y) = −∇ x2 δ(x − y) for model B conserved dynamics
[33]. Here the field φ can correspond to the fluctuations of a
range of order parameters in a lipid membrane, in the Gaussian
approximation, where the fluctuations are weak [20,21].
The order parameter in question could be compositional
fluctuations above the demixing transition in lipid bilayers
or binary fluids. Here a delta function form of the coupling K
would correspond to the tracer having a preference for one of
the lipid phases or preferential wetting for one of the phases
of a binary fluid. As well as compositional fluctuations, φ
could also correspond to fluctuations in local lipid ordering
(gel and liquid phases) and possibly orientational order in
the lipid tails. It could also correspond to local membrane
thickness in the case where the protein induces a local
hydrophobic mismatch and locally alters the thickness of the
bilayer.
To model a lipid membrane where the field represents the
height fluctuations and the particle is a protein coupled to
membrane curvature, we may take the Helfrich Hamiltonian
[34] (x, y) = (κ∇ x4 − σ ∇ x2 )δ(x − y), K(x, y) = −∇ x2 δ(x −
y), and R is given by its Fourier transform, R̃(k) = 1/4η|k|,
where η is the viscosity of the solvent surrounding the
membrane [35,36].
The computations below can be made a little more general.
As mentioned in the introduction there are a number of
physical cases of nonequilibrium systems where one is not
restricted to stochastic dynamics obeying detailed balance
and, for instance, the coupling between the particle and
the field may be taken as nonsymmetric: Instead of (5),
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PERTURBATIVE PATH-INTEGRAL STUDY OF ACTIVE- . . .
PHYSICAL REVIEW E 84, 011148 (2011)
we will consider
√
φ̇(x,t) = −κφ Rφ(x) + ζ hκφ RK[x − x(t)]+ κφ ξ (x,t),
where is a Gaussian noise dependent on the position in space
and time, with correlation function
(x,t)( y,s)T = T G(x − y,t − s),
(8)
(12)
and we have introduced the functions
where we have introduced the parameter ζ . The case of
stochastic dynamics with detailed balance is recovered for
ζ = 1. We note that Eq. (2) has been extensively studied in the
case where the field φ evolves independently of the particle
position, which corresponds to ζ = 0. This problem is referred
to as the advection diffusion of a passive scalar (the particle
concentration) in a fluctuating field φ. It had been suggested
that this form can be used to approximate the diffusion of an
active tracer particle in Refs. [16,18], for the case of a protein
weakly coupled to membrane curvature. In this approximation
it was found that the effect of the field fluctuations could be to
increase the diffusivity of the tracer particle with respect to that
obtained when it is not coupled to the fluctuating field (h = 0).
However, the numerical simulations of Ref. [13] where the
effect of the particle position on the field is taken into account
showed that the diffusion is reduced with respect to the case
h = 0, in agreement with later analytical studies [19,22].
Also as previously mentioned, a further generalization can
be made to our model: Since, in a nonequilibrium system, the
particle and the field are not necessarily driven by the same
thermal bath, they can experience different temperatures. In
this case the temperatures appearing in the correlation function
of the noises in Eqs. (3) and (6) will be denoted, respectively,
T x and Tφ .
F(x,u) = ζ h2 κ x κφ ∇Ke−κφ uR RK(x),
G(x,u) = −h2 κ x2 ∇∇ T K 2 e−κφ |u|R −1 (x).
We introduce the above functions for two reasons: to
provide more compact notation for the switch to the pathintegral formalism, and show explicitly which part of the
following computation is general and does not depend on
the expressions of F and G. Indeed, for a different choice
of functions, (11) and (12) define another model, which
may be analyzed using the method presented here. We will
need explicit expressions for F and G; in particular, their x
dependence is rather obscure. Fourier transforming allows us
to write them as a sum of functions with a completely explicit
x dependence:
dd k
˜
2
ike−κφ uR̃(k)(k) R̃(k)K̃ 2 (k)eik·x
F(x,u) = ζ h κ x κφ
d
(2π )
dd k
=
F k (x,u)
(15)
(2π )d
and
G(x,u) = h2 κ x2
=
III. EFFECTIVE DIFFUSION EQUATION AND
PATH-INTEGRAL FORMALISM
Our aim is to study the average value of the mean-squared
displacement of the particle, we thus integrate the dynamical
equation of field Eq. (8), assuming without loss of generality
that the field φ = 0 at time t = 0. This gives
φ(x,t) =
t
e−κφ (t−s)R
t
t
e−κφ (t−s)R {ζ hκφ RK[x(t) − x(s)]
√
√
+ κφ ξ [x(t),s]} ds + κ x η(t).
(10)
ẋ(t) = hκ x ∇K
−∞
The right-hand term can be split into two parts: a deterministic part depending only on the particle trajectory, and a
stochastic part that depends on the noise driving the field and
on the particle trajectory:
ẋ(t) =
√
κx η(t)
t
F[x(t) − x(s),t − s] ds + [x(t),t],
+
dd k
G k (x,u).
(2π )d
(16)
−∞
− (x(t),t) P [η]Q[][d x][dη][d],
(9)
Using this result in Eq. (2) we obtain the effective diffusion
equation for the particle:
˜
We now turn to the path-integral formalism; the first
steps are analogous to those described for general stochastic
dynamics in Refs. [37–39] and [40] for transport by a
time-dependent incompressible velocity field. The partition
function for this system is
t
√
Z=
δ ẋ(t) − κ x η(t) −
F[x(t) − x(s),t − s] ds
−∞
√
× {ζ hκφ RK[x − x(s)] + κφ ξ (x,s)} ds.
e−κφ |u|R̃(k)(k) K̃ 2 (k) ik·x
dd k
k kT
e
d
˜
(2π )
(k)
B. Path-integral formulation
A. Effective diffusion equation
(13)
(14)
(17)
where P and Q are the functional Gaussian weight for the
noises. Note that we should in principle include a Jacobian
in the above expression. However, if we use the Itô convention,
the Jacobian term is constant and independent of the path.
This is because the causality of the equation of motion and
the use of the Itô calculus mean that the transformation matrix
is triangular with diagonal terms that are constant [39]. We
then use a functional integral description of the δ function,
introducing the vector field p,
t
√
Z = exp i
p(t) · ẋ(t) − κ x η(t) −
F[x(t)
−∞
− x(s),t − s) ds − (x(t),t] dt
(11)
−∞
011148-3
× P [η]Q[][d x][d p][dη][d].
(18)
VINCENT DÉMERY AND DAVID S. DEAN
PHYSICAL REVIEW E 84, 011148 (2011)
Then we use the standard result that, for a Gaussian random
variable of zero mean u, exp(au) = exp(a 2 u2 /2) to perform the integration over the noises to obtain
Z=
exp(−S[x, p])[d x][d p],
(19)
where the action is given by
S[x, p] = −i
t
p(t) · ẋ(t) −
F[x(t)
−∞
− x(s),t − s] ds dt + D x | p(t)|2 dt
Tφ
+
pT (t)G(x(t) − x(s),t − s) p(s) dt ds. (20)
2
This action is the sum of the action of the pure Brownian
motion
p(t) · ẋ(t) dt + D x | p(t)|2 dt (21)
S0 [x, p] = −i
To obtain the two-point correlation functions, we use
the fact that the (functional) integral of a total (functional)
derivative is zero, for example,
δ
[ p(t)e−S0 ][d x][d p]
0=
δx(s)
(27)
=
p(t) ṗ(s)T e−S0 [d x][d p];
dividing each side by Z0 , we obtain p(t) ṗ(s)T 0 = 0:
p(t) p(s)T 0 is a constant. This constant must be zero, because
the action S0 does not correlate p at different times: p(t) are
all independent. We thus have the first correlator
p(t) p(s)T 0 = 0.
(28)
We use the same technique for the other correlators:
δ
[ p(t)e−S0 ][d x][d p]
0=
δ p(s)
= {δ(t − s)1 + p(t)[i ẋ(s)T − 2D x p(s)T ]}e−S0 [d x][d p],
(29)
and the action of the interaction
Sint [x, p] = i
p(t) · F[x(t) − x(s),t − s]θ (t − s) dt ds
Tφ
+
(22)
pT (t)G[x(t) − x(s),t − s] p(s) dt ds.
2
Since G(−x, −u) = G(x,u), we can write this integral only
with times satisfying t s, which will be convenient for the
ensuing calculations:
Sint [x, p]
=i
p(t) · F[x(t) − x(s),t − s]θ (t − s) dt ds
+ Tφ
pT (t)G[x(t) − x(s),t − s] p(s)θ (t − s) dt ds.
which gives, after dividing by Z0 and integrating over s ∈
[0,t],
x 0 (t) p(s)T 0 = iχ[0,t) (s)1,
(30)
where χA (s) is the characteristic function of the set A (equal
to 1 if the argument is in A and zero elsewhere). Finally the
identity,
δ
0=
(x 0 (t)e−S0 )[d x][d p]
δ p(s)
= x 0 (t)[i ẋ(s)T − 2D x p(s)T ]e−S0 [d x][d p], (31)
leads to
x 0 (t)x 0 (s)T 0 = 2D x L([0,t] ∩ [0,s])1,
(32)
(23)
where L(I ) is the length of the interval I ⊂ R; this is, of course,
the standard result for free Brownian motion x 0 (t)x 0 (s)T 0 =
2D x min(t,s) if t,s 0.
We will need to compute averages with the free action S0 .
Since it is quadratic in x and p, we need just the one- and
two-point correlation functions. Moreover, the position of the
particle is relevant only with respect to its position at, say,
t = 0. In order to keep compact notations, we define
IV. PERTURBATIVE CALCULATION OF THE EFFECTIVE
DIFFUSION CONSTANT
C. Computing averages with the free action
x 0 (t) = x(t) − x(0).
(24)
The correlation functions required are x 0 (t)0 , p(t)0 ,
p(t) p(s)T 0 , x 0 (t) p(s)T 0 , and x 0 (t)x 0 (s)T 0 .
Using the symmetry of S0 , we have immediately
x 0 (t)0 = 0,
p(t)0 = 0.
(25)
(26)
A. Derivation of the general result
Here we should go back to our aim of computing the
effective diffusion constant De . To do this, we have to evaluate
x 0 (tf )2 at a large time tf . Since we do not know how to
compute averages with the action S, we use a perturbation
expansion in terms of averages over S0 , which will be denoted
· · ·0 :
x 0 (tf )2 =
x 0 (tf )2 exp(−Sint [x, p])0
.
exp(−Sint [x, p])0
(33)
Averages appearing here are not easy to compute, but it is
easy to compute the first nontrivial term in expansion in
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PERTURBATIVE PATH-INTEGRAL STUDY OF ACTIVE- . . .
PHYSICAL REVIEW E 84, 011148 (2011)
the interaction action Sint . To do this, we just expand the
exponential functions:
x 0 (tf )2 (1 − Sint [x, p])0
.
1 − Sint [x, p]0
x 0 (tf )2 =e
− 12 kT x x T k
⎛
i
|J |
⎝
J ⊂N
k · Oj x
j ∈J
Sint,k [x, p]0 = 0.
(34)
The interaction action is linear in F and G and so are the
averages in (34); these functions are the sum over Fourier
modes of functions F k and G k , so we can carry out the
computation with only one Fourier mode and integrate over
all modes at the end. We denote by Sint,k the part of the action
due to the interaction with the k mode of the field and compute
Sint,k [x, p]0 and x 0 (tf )2 Sint,k [x, p]0 .
˜
In what follows we work at fixed wave vector k, so (k),
K̃(k), and R̃(k) are pure numbers, and to lighten the notation
we will write them , K, and R.
Every
average we have to compute is made of terms of the
form nj=1 Oj eik·x , where Oj are operators linear in x and
p. We will need the following formula, which is easy to derive
from Wick’s theorem,
n
Oj eik·x
j =1
path integral. For the same reason, p(t)T G k (x(t) − x(s),t −
s) p(s)0 = 0. Hence
⎞
Oj ⎠ , (35)
j ∈J
/
where N is the set {1, . . . ,n} and the sum over J denotes the
sum over all subsets of N .
We start with Sint,k [x, p]0 . This average contains two
integrated terms. The first is p(t) · F k [x(t) − x(s),t − s]0
with t > s, which involves
Now we turn to x 0 (tf )2 Sint,k [x, p]0 . We have to
and
x 0 (tf )2
compute
x 0 (tf )2 p(t)eik·[x(t)−x(s)] 0
T ik·[x(t)−x(s)]
p(t) p(s) e
0 , with s t. The only nonzero term in
the first average is
x 0 (tf )2 p(t)eik·[x(t)−x(s)] 0
= 2i p(t)x 0 (tf )T 0 x 0 (tf )[x(t) − x(s)]T 0 ke−k
= −4D x L([0,tf ] ∩ [s,t])ke−k
= −4D x [t − max(s,0)]k e
= i p(t)[x(t) − x(s)] 0 ke
−k 2 D x (t−s)
= 0,
D x (t−s)
−k D x (t−s)
2
× e−k
2
D x (t−s)
χ[0,tf ] (t).
(38)
(39)
0
=
and
2
2 2
4ζ h κ x κφ D x RK k
tf
(D x k 2 + κφ R)2
Tφ
x 0 (tf )2
2
(40)
p (t)G k [x(t) − x(s),t − s] p(s) dt ds
T
= 2Tφ h2 κ x2 −1 K 2
0
D x k − κφ Rk
tf .
(D x k 2 + κφ R)2
4
2
(41)
Thus, gathering these two results in (34) gives
κ x Tφ D x k 2 + (2ζ D x − κ x Tφ )κφ R
.
x 0 (tf )2 2dtf D x − h2 κ x k 2 K 2
d(D x k 2 + κφ R)2
Integrating over the modes, we get for the effective diffusion constant
2
˜
h2
dd k
2
2 κ x Tφ D x k + (2ζ D x − κ x Tφ )κφ R̃(k)(k)
De = D x −
K̃(k)
κ
k
.
x
d
2
2
˜
˜
d
(2π )
(k)[D x k + κφ R̃(k)(k)]
The integrals appearing in Eq. (43) will in some cases need
to be regularized by either introducing a large k (ultraviolet)
cutoff or a small k (infrared) cutoff. The former cutoff
χ[0,tf ] (t).
Now we just have to integrate the above results, and since
we are interested in the long-time behavior, we can neglect the
terms in o(tf ):
2
i x 0 (tf )
p(t) · F k [x(t) − x(s),t − s]θ (t − s) dt ds
(36)
B. Infrared behavior and the possibility of anomalous diffusion
χ[0,tf ] (t)
= {4D x [t − max(s,0)]k kT − 2χ[0,tf ] (s)1}
2
This expression is our main result. The large number of
parameters makes its interpretation quite difficult, so we will
apply it to some special cases to show the great variety of
phenomena that could be described.
D x (t−s)
x 0 (tf )2 p(t) p(s)T eik·[x(t)−x(s)] 0
with the notation k = |k| , and we have used that
p(t)x(t)T 0 = 0 because we use the Itô convention in our
2
2
2
The second contains two nonzero terms, and we get
p(t)eik·[x(t)−x(s)] 0
T
(37)
(42)
(43)
corresponds to the existence of a molecular length scale
below which the field φ cannot fluctuate. The latter case
is regularized by a length scale corresponding to the size
of the system. The dependence of the diffusion constant on
the system size is a sign of the onset of anomalous diffusion
as opposed to normal diffusion. To analyze when interaction
with the field maintains normal diffusion, we need to analyze
the problem in terms of the large distance behavior of the
problem. Physically a number of mechanisms can lead to
anomalous diffusion [1,41,42], and in potential driven systems
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VINCENT DÉMERY AND DAVID S. DEAN
PHYSICAL REVIEW E 84, 011148 (2011)
subdiffusion can be induced by a divergence in the average time
to cross energy barriers. While subdiffusion can be induced by
diverging time scales, it can also be induced by the presence
of long-range correlations or diverging length scales. While
in potential problems trapping in local minima can lead to
subdiffusion, long-range correlations in incompressible fields
can lead to superdiffusion.
We proceed as in Refs. [20,21] by defining the following
exponents associated with the small k behavior of the operators
appearing in the problem:
˜
(44)
(k)
∼ kδ ,
K̃(k) ∼ k α
(45)
R̃(k) ∼ k ρ .
(46)
Now, from Eq. (43) we see that there are two distinct regimes
that control the small k behavior of the integrals appearing in
it. The regimes are ρ + δ < 2 and ρ + δ > 2, and physically
we can understand this difference between these regimes. It
suffices to note that the free field has a time-dependent length
1
scale lφ (t) ∼ t ρ+δ , whereas the bare diffusion has a length
1
scale l x (t) ∼ t 2 . The regime where ρ + δ < 2 corresponds to
the case where l x (t) < lφ (t) and vice versa.
The first, which we refer to as the adiabatic regime, is
where ρ + δ < 2. This means that the integrals are dominated
˜ rather than the terms D x k 2 , which
at low k by the terms R̃ means that we effectively find ourselves in the adiabatic limit
where κ x κφ as defined in Refs. [19–21], which physically
corresponds to field dynamics that is much quicker than the
bare diffusion of the tracer. In this case the field is in a local
equilibrium about the tracer, a fact that is seen in the adiabatic
path-integral approach of Ref. [19]. In this case there is a
critical dimension dc given by
dc = ρ − 2α + 2δ − 2
(47)
such that for d > dc the diffusion is normal. For d < dc the
diffusion will be anormal. There is a second possibility where
ρ + δ > 2. In this case the bare diffusion of the particle is
more rapid than the field dynamics, and we are in the opposite
limit to the adiabatic one. Here we find that
dc = δ − 2α.
(48)
C. Application to some special cases
1. Stochastic dynamics with detailed balance
We first analyze the case where the particle and the field see
the same temperature, T x = Tφ = T , and the dynamics obeys
detailed balance, i.e., ζ = 1. The effective diffusion constant
is thus
h2
κ x k 2 K̃(k)2
dd k
.
Dedb = D x 1 −
2
˜
˜
d
(2π )d (k)[D
x k + κφ R̃(k)(k)]
not just in the regime of small h can be shown explicitly
within the Kubo formalism [22]. A physical explanation for
the slowing of diffusion in this case can be found in studies of
the drag on a particle coupled to a field. The reaction of the
field to the particle is to create a polaron-like deformation of
the field about the particle; however, a moving particle has a
polaron that is not symmetric with respect to the front and rear
of the particle. This deformation generates a drag force that
tends to pull the particle backward [20,21]. The above result
also agrees, to O(h2 ) in perturbation theory, with studies of
the drag on proteins in membranes that couple to membrane
curvature [13,19,22] in the adiabaitc limit defined above.
These studies analyzed the adiabatic limit via, respectively,
phenomonological arguments, a saddle-point approximation in
the path-integral formulation, and an operator formalism. The
former two studies analyzed the case of quadratic couplings,
but in the limits where the coupling becomes linear the results
agree with those obtained here and in Ref. [22]. These results
are obtained from Eq. (49) by taking the adiabatic limit
D x → 0 in the denominator of the integral.
2. Passive diffusion
For passive diffusion, that is, ζ = 0, and still with equal
temperatures for the particle and field, we have
h2
dd k
pass
κ x k 2 K̃(k)2
De = D x 1 −
d
(2π )d
˜
D x k 2 − κφ R̃(k)(k)
.
(50)
×
2
2
˜
˜
(k)[D
x k + κφ R̃(k)(k)]
This result shows that, depending on the speeds of evolution κ x
and κφ of the particle and the field, the particle’s diffusion may
increase or decrease. This is in agreement with the following
intuition: With a slow field we are close to diffusion in a
quenched potential, which slows the diffusion due to trapping
in local minima that are temporally persistent. However, when
the field fluctuates quickly, the field fluctuations kick the
particle along, thus adding to the effective random force
the particle experiences. However, this picture is not totally
valid: There appears an optimal value of κφ at which the
perturbative enhancement of the tracer’s diffusion constant
is maximal. If the field fluctuates too quickly, the effect of
field fluctuations simply averages out to zero. This result
is quite difficult to understand physically, but it resembles
closely the phenomenon of stochastic resonance [29–31],
where the application of a periodic but deterministic potential
to Brownian particles can show an optimal frequency at which
the particle dispersion is maximized. Here the optimal value
of κφ depends, in particular, on the bare diffusivity κ x of the
particle; thus two different species will react differently to the
same external field. We note that this type of phenomenon can
be used to sort molecules [32].
(49)
This result is exactly what was found in Ref. [22] via a
Kubo formula formalism that we emphasize applies only to
this particular case. By inspecting Eq. (49) we see that the
correction to the bare diffusion constant is always negative,
and the diffusion is thus slowed by its coupling to the field.
The fact that the diffusion is slowed for all values of h and
3. Particle not connected to a thermal bath
Another case which could have physical relevance is where
the particle is not connected to a thermal bath, i.e., T x = 0,
and energy enters the system only via the fluctuations of
the field. Hence in the absence of coupling to the field, the
particle cannot diffuse, as D x = 0. Our result shows that
011148-6
PERTURBATIVE PATH-INTEGRAL STUDY OF ACTIVE- . . .
PHYSICAL REVIEW E 84, 011148 (2011)
the fluctuations of the field will induce a nonzero diffusion
constant for the particle:
h2
d d k κ x2 k 2 K̃(k)2
.
(51)
DeTx =0 = Tφ
˜ 2
d
(2π )d κφ R̃(k)(k)
0.8
De
The effective diffusion constant is now proportional to the
temperature seen by the field Tφ : In some sense, the field acts
as a thermal bath for the particle. Note that the effect of field
fluctuations is to speed up the diffusion from a zero diffusion
constant to that given by Eq. (51); this is in agreement with
the physical intuition that the field fluctuations will help the
particle to disperse. Interestingly, we see that when D x = 0,
the result for De is independent of ζ, and the active and passive
cases have the same diffusion constant.
1
0.4
0
V. NUMERICAL SIMULATIONS
In this section we test out our theoretical predictions against
the numerical simulations of a toy model in one dimension.
We should bear in mind that the simulation of the diffusion of
active tracers is more computationally intensive than that for
passive tracers. In the latter case we can simulate the diffusion
of an ensemble of independent tracers on a given dynamical
realization of the fluctuating field. In the former case, however,
we must follow the diffusion of a single particle in the
fluctuating field as for a system with more than one particle
the coupling to the field introduces interactions between
the particles [23,24]. In what follows we will numerically
compute the evolution of Eq. (5) in Fourier space and while
integrating Eq. (2) in real space to generate single-particle
trajectories, which are then ensemble averaged to estimate the
diffusion constant.
A. Numerical model
We consider the simplest model for our numerical simulations, d = 1, and we take a finite number of modes, with
˜
k = ±n, 1 n N. We also take (k)
= k 2 , K̃(k) = 1,
and R̃(k) = 1. We set Tφ = 1, κ x = 1. For this choice of
parameters, our result [Eq. (43)] reads
De = D x − h
2 Dx
+ κφ (2ζ D x − 1) 1
.
π (D x + κφ )2
n2
n=1
0.6
0.5
1
h
1.5
2
FIG. 1. Effective diffusion coefficient for stochastic dynamics
with detailed balance (Gaussian ferromagnet with model A dynamics:
˜ = k 2 , R̃ = 1, K̃ = 1) with a single-mode field as a function of the
coupling constant h. The crosses represent numerical simulations; the
solid line is the perturbative result Eq. (52).
B. Validity range of the perturbative result
The first question that arises is: To what extent is our
perturbative result valid? To find the validity range for h, we
take stochastic dynamics with detailed balance, with one mode
and κφ = 1, and look at the effective diffusion coefficient as a
function of h. The comparison between the simulations and the
result (52) is given Fig. 1, and it shows that our computation
is valid (i.e., the relative error is less than 5%) for h 1.2. A
more relevant criterion is, however, to what extent the deviation
from the bare result can be predicted by our result. Figure 1
shows that the theory predicts the deviations of the diffusion
constant from its bare value in the region where the diffusion
constant deviates of the order of 15% –20% from its bare
value.
C. Stochastic dynamics with detailed balance
N
(52)
We will perform four simulations: First, we consider
stochastic dynamics with detailed balance, and let the coupling h vary, to explore the range of validity of our perturbative
result. Then we will simulate the three special cases described
above, for different values of the speed of evolution of the
field κφ . Each simulation is performed for one mode and 10
modes.
In the simulations, we let one particle evolve in the field for a
long time τ κ x−1 , κφ−1 , and we measure its position at a fixed
set of times. We repeat this simulation a large number of times
(around 105 ) and, using these measurements, we compute
x(t)2 /2dt, where t is the measurement time. For large t,
this function fluctuates around a mean value, which gives us
the effective diffusion constant. When these fluctuations are
not small, they are taken into account with error bars on the
plots.
We can also vary the rate of evolution of the field κφ . We
set D x = 1 and plot Dedb (κφ ) for one mode and h = 1, and for
10 modes and h = 0.5, comparing the numerical simulations
results with (52). The results are shown in Fig. 2. For one
mode and h = 1, at the border of the range of validity of the
perturbative approach, our results are in quite good agreement
with the simulations.
D. Passive diffusion
For passive diffusion, the results of numerical simulations
are shown Fig. 3 and compared to Eq. (52). We see that
the analytical predictions are in good agreement with the
results of simulations. In particular we see that depending
on the relative values of κ x and κφ , the diffusion is either
slowed down or speeded up. For small values of κφ the
diffusion is reduced; however, on increasing κφ the diffusion
speeds up and passes through a maximum before decaying
011148-7
VINCENT DÉMERY AND DAVID S. DEAN
PHYSICAL REVIEW E 84, 011148 (2011)
(a)
(b)
1
0.9
De
0.95
0.8
0.9
0.7
0
2.5
κφ
5
0
2.5
κφ
5
FIG. 2. Effective diffusion coefficient for stochastic dynamics with detailed balance (Gaussian ferromagnet with model A dynamics:
˜ = k 2 , R̃ = 1, K̃ = 1) as a function of κφ with D x = 1 and (a) 1 mode and h = 1 (b) 10 modes and h = 0.5: numerical simulations (crosses)
and perturbative results (solid lines).
toward the bare value D x = 1 as predicted by our perturbative
calculations.
E. Particle not connected to any thermal bath
The results for numerical simulations when the particle is
not connected to any thermal bath are shown Fig. 4. They
are in good agreement with Eq. (52) for κφ 1.5. When
κφ → 0, according to our computation, the effective diffusion
coefficient diverges, whereas physically it should go to zero,
which is confirmed by the simulations. This discrepancy comes
from the fact that we neglected the terms in o(tf ) in our
computation of x(tf )2 Sint 0 , and we took the limit tf → ∞
before taking the limit κφ → 0.
VI. DIFFUSION COEFFICIENT FROM THE PROBABILITY
DENSITY FOR THE PASSIVE CASE
is described in Refs. [1,2]. In terms of the general model of
this paper we are thus considering the special case ζ = 0 and
T x = Tφ = T .
First, we give the elementary equations for the pure
Brownian motion. The probability density function P0 (x,t)
of a particle starting at x = 0 when t = 0, setting P0 (x,t <
0) = 0, satisfies
Ṗ0 (x,t) = ∇ · [D x ∇P (x,t)] + δ(x)δ(t).
(53)
We can write this equation in terms the free diffusion operator
H0 = ∂t − ∇ · (D x ∇):
H0 P0 (x,t) = δ(x)δ(t),
(54)
and Fourier transform P̃0 (k,ω) = d x dtP0 (x,t)e−i(k·x+ωt) is
given by
Here we use a perturbation expansion of the Fokker-Planck
equation in order to compute the perturbative correction for
passive diffusion in a fluctuating field. The basic formalism
P̃0 (k,ω) =
(a)
1
.
D x k 2 + iω
(55)
(b)
1
De
1
0.95
0.9
0.9
0.8
0
2.5
κφ
5
0
2.5
κφ
5
˜ = k 2 , R̃ = 1, K̃ = 1) as a
FIG. 3. Effective diffusion coefficient for passive diffusion (Gaussian ferromagnet with model A dynamics: function of κφ with D x = 1 and (a) 1 mode and h = 1 (b) 10 modes and h = 0.5: numerical simulations (crosses) and perturbative results
(solid lines).
011148-8
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PHYSICAL REVIEW E 84, 011148 (2011)
(b)
(a)
0.06
0.075
0.04
De
0.05
0.02
0.025
0
1
2
0
3
1
κφ
2
3
κφ
FIG. 4. Effective diffusion coefficient for a particle not connected to any thermal bath (Gaussian ferromagnet with model A dynamics:
˜ = k 2 , R̃ = 1, K̃ = 1) as a function of κφ with κ x = 1 and (a) 1 mode and h = 0.3, (b) 10 modes and h = 0.3: numerical simulations (crosses)
and perturbative results (solid lines).
The effective diffusion coefficient De of a process can thus be
extracted from the Fourier transform of its probability density
function P̃ (k,ω) via [1,2]
−1
(56)
D = lim k 2 P̃ (k,0) .
so P̃ (k,ω) is given by the integral equation:
dq dν
k·q
P̃ (k,ω) = P̃0 (k,ω) 1 − hκ x
(2π )d+1
× K̃(q)φ̃(q,ν)P̃ (k − q,ω − ν) .
|k|→0
We are indeed interested in the large distance behavior of
our system, which is why the effective diffusion coefficient is
given by the small wave-vector behavior. With this equation in
hand, our strategy is simple: Compute the probability density
function for the passive diffusion and extract the effective
diffusion coefficient.
Now, we introduce a given field φ(x,t). For the general
model described above, Eq. (2) gives the diffusion operator,
which replaces H0 in Eq. (54):
H = H0 + Hint = ∂t − ∇ · {D x ∇ + hκ x [∇(Kφ)(x,t)]}.
(57)
In this equation, the probability density function is that of
pure Brownian motion, perturbed to the order h. Iterating this
equation gives an explicit expression of P̃ (k,ω) up to the
desired order of h.
Once we have the equation for P̃ (k,ω) for a given field we
need to extract the effective diffusion coefficient and proceed
by averaging (59) over the configurations of the field φ(x,t)
(which does not depend on the particle position). The field
has a Gaussian probability density function with a two-point
correlation function that can easily be computed from Eq. (9):
φ̃(q,ν)φ̃(q ,ν )
=
The equation satisfied by the Fourier transform P̃ (k,ω)
is
H
P (k,ω) = (iω + D x k 2 )P̃ (k,ω)
dq dν
+ hκ x
k · q K̃(q)φ̃(q,ν)
(2π )d+1
× P̃ (k − q,ω − ν) = 1,
(58)
P̃ (k,ω) = P̃0 (k,ω) 1 − 2h2 P̃0 (k,ω)T κφ κ x2
2T κφ R̃(q)
(2π )d+1 δ(q + q )δ(ν + ν ).
2
˜
ν 2 + [κφ R̃(q)(q)]
(60)
Now we have to insert this average in an explicit perturbative expansion of P̃ (k,ω) given by (59). The lowest nonzero
order is the second order: The field φ(x,t) has to appear at
least twice. Moreover, we will obtain the probability density
function to the order h2 , which is exactly what we did with the
path-integral method. Computing P̃ (k,ω) to the order h2 and
averaging the field out leads to
dq dν k · q (k − q) · q R̃(q)K̃(q)2
P̃
(k
−
q,ω
−
ν)
.
0
2
˜
(2π )d+1
ν 2 + [κφ R̃(q)(q)]
Then we can restrict ourselves to ω = 0, use the expression (55) and integrate over ν, using
P̃ (k,0) =
(59)
dν
1
2π (iν+α)(ν 2 +β 2 )
h2 κ x
k · q (k − q) · q K̃(q)2
dq
1
.
1
−
2
˜
˜
Dx k2
k2
(2π )d (q)[D
x (k − q) + κφ R̃(q)(q)]
011148-9
=
(61)
1
:
2β(α+β)
(62)
VINCENT DÉMERY AND DAVID S. DEAN
PHYSICAL REVIEW E 84, 011148 (2011)
Finally, we just need to determine to behavior of the above expression when |k| → 0; a straightforward computation gives
˜
h2 κ x
1
dq q 2 K̃(q)2 [D x q 2 − κφ R̃(q)(q)]
P̃ (k,0) ∼
1
+
,
(63)
2
2
˜
˜
|K |→0 D x k 2
d
(2π )d (q)[D
x q + κφ R̃(q)(q)]
and we recover the effective diffusion coefficient given in (50):
˜
h2 κ x
D x q 2 − κφ R̃(q)(q)
dd k 2
2
Depass = D x 1 −
.
K̃(q)
q
2
2
˜
˜
d
(2π )d
(q)[D
x q + κφ R̃(q)(q)]
VII. CONCLUSIONS
We have analyzed the diffusive behavior of a tracer particle
diffusing in a time-dependent Gaussian potential in the limit of
weak coupling between the particle and the field. The method
has the advantage that it can be applied to a wide range of
models and not just the cases of passive diffusion or active
diffusion with detailed balance. In these two aforementioned
cases the method agrees with results obtained, respectively,
via a perturbation expansion of the Fokker Planck equation
and a Kubo formulation. We have also been able to look at
nonequilibrium systems with variable feedback of the tracer
on the field and systems where the field and the tracer are
subject to thermal noise of different temperatures. The range
of behavior seen in the late-time diffusion coefficient is quite
rich, and depending on the models considered, coupling to
the field can either slow down or speed up the diffusion. The
speeding up or slowing down of diffusion and the possibility
of a form of stochastic resonance depends on the relative
rates of the dynamics of the fluctuating field and the bare
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